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Journal articles on the topic 'Integer partition theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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1

GARVAN, FRANK G., and HAMZA YESILYURT. "SHIFTED AND SHIFTLESS PARTITION IDENTITIES II." International Journal of Number Theory 03, no. 01 (March 2007): 43–84. http://dx.doi.org/10.1142/s1793042107000808.

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Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

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2

Yan, Xiao-Hui. "Partitions of the set of nonnegative integers with identical representation functions." International Journal of Number Theory 15, no. 10 (November 2019): 1969–75. http://dx.doi.org/10.1142/s1793042119501070.

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Let [Formula: see text] be the set of nonnegative integers. For any set [Formula: see text], let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] with [Formula: see text]. Chen and Wang proved that the set of positive integers can be partitioned into two subsets [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. In this paper, we prove that, for a given integer [Formula: see text] and a partition [Formula: see text], there is an integer [Formula: see text] such that [Formula: see text] does not hold.

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3

RØDSETH, ØYSTEIN J., and JAMES A. SELLERS. "PARTITIONS WITH PARTS IN A FINITE SET." International Journal of Number Theory 02, no. 03 (September 2006): 455–68. http://dx.doi.org/10.1142/s1793042106000644.

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For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

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4

Ballantine, Cristina, and Mircea Merca. "Combinatorial proof of the minimal excludant theorem." International Journal of Number Theory 17, no. 08 (February 26, 2021): 1765–79. http://dx.doi.org/10.1142/s1793042121500615.

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The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new combinatorial interpretation for [Formula: see text]. They showed, using generating functions, that [Formula: see text] equals the number of partitions of [Formula: see text] into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function [Formula: see text]. We generalize this combinatorial interpretation to [Formula: see text], the sum of least [Formula: see text]-gaps in all partitions of [Formula: see text]. The least [Formula: see text]-gap of a partition [Formula: see text] is the smallest positive integer that does not appear at least [Formula: see text] times as a part of [Formula: see text].

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5

BRENNAN, CHARLOTTE, ARNOLD KNOPFMACHER, and STEPHAN WAGNER. "The Distribution of Ascents of Size d or More in Partitions of n." Combinatorics, Probability and Computing 17, no. 4 (July 2008): 495–509. http://dx.doi.org/10.1017/s0963548308009073.

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A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

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6

Andrews, George E. "The Bhargava-Adiga Summation and Partitions." Journal of the Indian Mathematical Society 84, no. 3-4 (July 1, 2017): 151. http://dx.doi.org/10.18311/jims/2017/15836.

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The Bhargava-Adiga summation rivals the 1ψ1􀀀summation of Ramanujan in elegance. This paper is devoted to two applications in the theory of integer partitions leading to partition questions related to Gauss's celebrated three triangle theorem.

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7

Calkin, Neil, Jimena Davis, Kevin James, Elizabeth Perez, and Charles Swannack. "Computing the integer partition function." Mathematics of Computation 76, no. 259 (February 28, 2007): 1619–39. http://dx.doi.org/10.1090/s0025-5718-07-01966-7.

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8

KAAVYA, S. J. "CRANK 0 PARTITIONS AND THE PARITY OF THE PARTITION FUNCTION." International Journal of Number Theory 07, no. 03 (May 2011): 793–801. http://dx.doi.org/10.1142/s1793042111004381.

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A well-known problem regarding the integer partition function p(n) is the parity problem, how often is p(n) even or odd? Motivated by this problem, we obtain the following results: (1) A generating function for the number of crank 0 partitions of n. (2) An involution on the crank 0 partitions whose fixed points are called invariant partitions. We then derive a generating function for the number of invariant partitions. (3) A generating function for the number of self-conjugate rank 0 partitions.

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9

Svaiter, B. F., and N. F. Svaiter. "The distributional zeta-function in disordered field theory." International Journal of Modern Physics A 31, no. 25 (September 8, 2016): 1650144. http://dx.doi.org/10.1142/s0217751x1650144x.

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In this paper, we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which cannot be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered [Formula: see text] model defined in a [Formula: see text]-dimensional Euclidean space.

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10

PENNISTON, DAVID. "ARITHMETIC OF ℓ-REGULAR PARTITION FUNCTIONS." International Journal of Number Theory 04, no. 02 (April 2008): 295–302. http://dx.doi.org/10.1142/s1793042108001341.

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Let bℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of bℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

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11

Huang, H. Y., and F. Y. Wu. "The Infinite-State Potts Model and Solid Partitions of an Integer." International Journal of Modern Physics B 11, no. 01n02 (January 20, 1997): 121–26. http://dx.doi.org/10.1142/s0217979297000150.

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It has been established that the infinite-state Potts model in d dimensions generates restricted partitions of integers in d-1 dimensions, the latter a well-known intractable problem in number theory for d>3. Here we consider the d=4 problem. We consider a Potts model on an L × M × N × P hypercubic lattice whose partition function GLMNP(t) generates restricted solid partitions on an L × M × N lattice with each part no greater than P. Closed-form expressions are obtained for G222P(t) and we evaluated its zeroes in the complex t plane for different values of P. On the basis of our numerical results we conjecture that all zeroes of the enumeration generating function GLMNP(t) lie on the unit circle |t|=1 in the limit that any of the indices L, M, N, P becomes infinite.

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12

Eggleton. "Equisum Partitions of Sets of Positive Integers." Algorithms 12, no. 8 (August 11, 2019): 164. http://dx.doi.org/10.3390/a12080164.

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Let V be a finite set of positive integers with sum equal to a multiple of the integer b. When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V, where n is a given positive integer: (1) an initial interval a∈Z+:a≤n; (2) an initial interval of primes p∈P:p≤n, where P is the set of primes; (3) a divisor set d∈Z+:d|n; (4) an aliquot set d∈Z+:d|n, d<n. Open general questions and conjectures are included for each of these classes.

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13

Acosta Diaz, Róbinson J., Christian D. Rodríguez-Camargo, and Nami F. Svaiter. "Directed Polymers and Interfaces in Disordered Media." Polymers 12, no. 5 (May 6, 2020): 1066. http://dx.doi.org/10.3390/polym12051066.

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We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contribute. The structure of field space is analysed for polymers and interfaces at finite temperature using the saddle-point equations derived from each integer moments of the partition function. For the case of an interface we obtain the wandering exponent ξ = ( 4 − d ) / 2 , also obtained by the conventional replica method for the replica symmetric scenario.

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14

MILAS, ANTUN, ERIC MORTENSON, and KEN ONO. "NUMBER THEORETIC PROPERTIES OF WRONSKIANS OF ANDREWS–GORDON SERIES." International Journal of Number Theory 04, no. 02 (April 2008): 323–37. http://dx.doi.org/10.1142/s1793042108001377.

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For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews–Gordon q-series [Formula: see text] This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of [Formula: see text] Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ±a ( mod b), then for every positive integer n, we have [Formula: see text] We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k + 1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p.

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15

Safriadi, Safriadi, Hasmawati Hasmawati, and Loeky Haryanto. "Partition Dimension of Complete Multipartite Graph." Jurnal Matematika, Statistika dan Komputasi 16, no. 3 (April 28, 2020): 365. http://dx.doi.org/10.20956/jmsk.v16i3.7278.

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Determining a resolving partition of a graph is an interesting study in graph theory due to many applications like censor design, compound classification in chemistry, robotic navigation and internet network. Let and , the distance between an is . For an ordered partition of , the representation of with respect to is . The partition is called a resolving partition of if all representation of vertices are distinct. The partition dimension of graph is the smallest integer such that has a resolving partition with element.In this thesis, we determine the partition dimension of complete multipartite graph , which is limited by , with and . We found that , , and , .

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16

Christopher, A. David. "Remainder sum and quotient sum function." Discrete Mathematics, Algorithms and Applications 07, no. 01 (February 2, 2015): 1550001. http://dx.doi.org/10.1142/s1793830915500019.

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This paper is concerned with two arithmetical functions namely remainder sum function and quotient sum function which are respectively the sequences A004125 and A006218 in Online Encyclopedia of Integer Sequences. The remainder sum function is defined by [Formula: see text] for every positive integer n, and quotient sum function is defined by [Formula: see text] where q(n, i) is the quotient obtained when n is divided by i. We establish few divisibility properties these functions enjoy and we found their bounds. Furthermore, we define restricted remainder sum function by RA(n) = ∑k∈A n mod k where A is a set of positive integers and we define restricted quotient sum function by QA(n) = ∑k∈A q(n, k). The function QA(n) is found to be a quasi-polynomial of degree one when A is a finite set of positive integers and RA(n) is found to be a periodic function with period ∏a∈A a. Finally, the above defined four functions found to have recurrence relation whose derivation requires few results from integer partition theory.

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17

Song, Jijian, and Bin Xu. "On Rational Functions with More than Three Branch Points." Algebra Colloquium 27, no. 02 (May 7, 2020): 231–46. http://dx.doi.org/10.1142/s100538672000019x.

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Let d be a positive integer and Λ be a collection of partitions of d of the form (a1, …, ap), (b1, …, bq), (m1 + 1, 1, …, 1), …, (ml + 1, 1, …, 1), where (m1, …, ml) is a partition of p + q − 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Λ if and only if max(m1, …, ml) < d/GCD(a1, …, ap, b1, …, bq). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

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18

KIM, BYUNGCHAN. "A REMARK ON TAIL DISTRIBUTIONS OF PARTITION RANK AND CRANK." Bulletin of the Australian Mathematical Society 93, no. 1 (August 20, 2015): 31–36. http://dx.doi.org/10.1017/s000497271500088x.

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We examine the tail distributions of integer partition ranks and cranks by investigating tail moments, which are analogous to the positive moments introduced by Andrews et al. [‘The odd moments of ranks and cranks’, J. Combin. Theory Ser. A120(1) (2013), 77–91].

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19

Mutafchiev, Ljuben R. "On the Maximal Multiplicity of Parts in a Random Integer Partition." Ramanujan Journal 9, no. 3 (June 2005): 305–16. http://dx.doi.org/10.1007/s11139-005-1870-9.

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20

Du, Julia Q. D., Edward Y. S. Liu, and Jack C. D. Zhao. "Congruence properties of pk(n)." International Journal of Number Theory 15, no. 06 (July 2019): 1267–90. http://dx.doi.org/10.1142/s1793042119500714.

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We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

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21

CARNEY, ALEXANDER, ANASTASSIA ETROPOLSKI, and SARAH PITMAN. "POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS." International Journal of Number Theory 08, no. 03 (April 7, 2012): 599–611. http://dx.doi.org/10.1142/s1793042112500339.

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Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

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22

Penniston, David. "11-Regular partitions and a Hecke eigenform." International Journal of Number Theory 15, no. 06 (July 2019): 1251–59. http://dx.doi.org/10.1142/s1793042119500696.

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A partition of a positive integer [Formula: see text] is called [Formula: see text]-regular if none of its parts is divisible by [Formula: see text]. Let [Formula: see text] denote the number of 11-regular partitions of [Formula: see text]. In this paper we give a complete description of the behavior of [Formula: see text] modulo [Formula: see text] when [Formula: see text] in terms of the arithmetic of the ring [Formula: see text]. This description is obtained by relating the generating function for these values of [Formula: see text] to a Hecke eigenform, and as a byproduct we find exact criteria for which of these values are divisible by 5 in terms of the prime factorization of [Formula: see text].

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23

Goh, William M. Y., and Eric Schmutz. "The number of distinct part sizes in a random integer partition." Journal of Combinatorial Theory, Series A 69, no. 1 (January 1995): 149–58. http://dx.doi.org/10.1016/0097-3165(95)90111-6.

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24

Sapucaia, Allan, Pedro J. de Rezende, and Cid C. de Souza. "Solving the minimum convex partition of point sets with integer programming." Computational Geometry 99 (December 2021): 101794. http://dx.doi.org/10.1016/j.comgeo.2021.101794.

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25

Holesovky, Jan. "Distribution Endpoint Estimation Assessment for the use in Metaheuristic Optimization Procedure." MENDEL 24, no. 1 (June 1, 2018): 93–100. http://dx.doi.org/10.13164/mendel.2018.1.093.

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Metaheuristic algorithms are often applied to numerous optimization problems, involving large-scale and mixed-integer instances, specifically. In this contribution we discuss some refinements from the extreme value theory to the lately proposed modification of partition-based random search. The partition-based approach performs iterative random sampling at given feasible subspaces in order to exclude the less favourable regions. The quality of particular regions is evaluated according to the promising index of a region. From statistical perspective, determining the promising index is equivalent to the endpoint estimation of a probability distribution induced by the objective function at the sampling subspace. In the following paper, we give a short review of the recent endpoint estimators derived on the basis of extreme value theory, and compare them by simulations. We discuss also the difficulties in their application and suitability of the estimators for various optimization instances.

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26

GARVAN, F. G. "CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK." International Journal of Number Theory 06, no. 02 (March 2010): 281–309. http://dx.doi.org/10.1142/s179304211000296x.

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Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

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DAMGAARD, P. H., and J. LACKI. "PARTITION FUNCTION ZEROS OF AN ISING SPIN GLASS." International Journal of Modern Physics C 06, no. 06 (December 1996): 819–43. http://dx.doi.org/10.1142/s012918319500068x.

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We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization group flows in such models with frustration, and by the possible connection of the latter with spin glass behavior. In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class of random polynomials with positive integer coefficients. Some aspects of this problem have been studied in various areas of mathematics, and we show in particular how polynomial mappings which are used in graph theory to classify graphs, may help in characterizing the distribution of zeros. We finally discuss the possible limiting set of these zeros as the volume is sent to infinity.

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Chen, Qi, Minquan Cheng, and Baoming Bai. "Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding." Entropy 23, no. 3 (March 9, 2021): 323. http://dx.doi.org/10.3390/e23030323.

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In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the (M,v) almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order n≤5 with the exception of log10·U2,5 and logv·U3,5 for some v.

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DAS, SUMIT R., AVINASH DHAR, GAUTAM MANDAL, and SPENTA R. WADIA. "W-INFINITY WARD IDENTITIES AND CORRELATION FUNCTIONS IN THE c=1 MATRIX MODEL." Modern Physics Letters A 07, no. 11 (April 10, 1992): 937–53. http://dx.doi.org/10.1142/s0217732392000835.

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We explore consequences of W-infinity symmetry in the fermionic field theory of the c=1 matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a three-dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two-point function of the bilocal operator in the double scaling limit. We extract the operator whose two-point correlator has a single pole at an (imaginary) integer value of the energy. We then rewrite the W-infinity charges in terms of operators in the matrix model and use this to derive constraints satisfied by the partition function of the matrix model with a general time dependent potential.

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KANEKO, ATSUSHI, and MIKIO KANO. "SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS." International Journal of Computational Geometry & Applications 15, no. 03 (June 2005): 229–38. http://dx.doi.org/10.1142/s0218195905001671.

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Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m-1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪⋯∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi∩B| = m-1 if |Xi∩R1| = 1, and |Xi∩B| = m if |Xi∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balanced partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1,…,Tg be g disjoint rooted trees such that |Ti| ∈ {m,m+1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+⋯+|Tg| points in the plane in general position that contains g specified points p1,…,pg. Then the rooted forest T1 ∪ ⋯ ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.

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Pittel, B. "Limit shape of a random integer partition with a bounded max-to-min ratio of parts sizes." Journal of Combinatorial Theory, Series A 114, no. 7 (October 2007): 1238–53. http://dx.doi.org/10.1016/j.jcta.2007.01.006.

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SOBERÓN, PABLO. "Robust Tverberg and Colourful Carathéodory Results via Random Choice." Combinatorics, Probability and Computing 27, no. 3 (December 19, 2017): 427–40. http://dx.doi.org/10.1017/s0963548317000591.

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We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

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VECCHIA, P. DI, M. KATO, and N. OHTA. "DOUBLE SCALING LIMIT IN O(N) VECTOR MODELS IN D DIMENSIONS." International Journal of Modern Physics A 07, no. 07 (March 20, 1992): 1391–413. http://dx.doi.org/10.1142/s0217751x92000612.

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Using the standard 1/N expansion, we study O(N) vector models in D dimensions with an arbitrary potential. We limit ourselves to renormalizable theories. We show that there exists a value of the coupling constant corresponding to a critical point and that a double scaling limit can be performed as in D=0 and in the case of matrix models in D=0, 1. For D=1 the theory is renormalizable with an arbitrary potential and we find in general a hierarchy of critical theories labeled by an integer k. The universal partition function obtained in the double scaling limit is constructed. Finally, we show that the critical behaviour of those models is the same as a branched polymer model recently constructed by Ambjørn, Durhuus and Jónsson.

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XU, BAOGANG, and XINGXING YU. "Better Bounds for k-Partitions of Graphs." Combinatorics, Probability and Computing 20, no. 4 (May 31, 2011): 631–40. http://dx.doi.org/10.1017/s0963548311000204.

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Let G be a graph with m edges, and let k be a positive integer. We show that V(G) admits a k-partition V1, . . . Vk such that $e(V_i)\leq \frac 1{k^2}m+\frac {k-1}{2k^2}(\sqrt{2m+1/4}-1/2)$ for i ∈ {1, 2, . . . k}, and $e(V_1, \ldots, V_k)\geq \frac{k-1}{ k} m +\frac{k-1}{ 2k}\sqrt{2m+1/4} +O(k)$, where e(Vi) denotes the number of edges with both ends in Vi and $e(V_1,\ldots, V_k)=m-\sum_{i=1}^ke(V_i)$. This answers a problem of Bollobás and Scott [2] in the affirmative. Moreover, $\binom{k+1}{ 2}e(V_i)+\frac k2\sum_{j\ne i}e(V_j)\le m + O(k^2)$ for i ∈ {1, 2, . . ., k}, which is close to being best possible and settles another problem of Bollobás and Scott [2].

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35

Matsoukas, Themis. "Statistical Mechanics of Discrete Multicomponent Fragmentation." Condensed Matter 5, no. 4 (October 18, 2020): 64. http://dx.doi.org/10.3390/condmat5040064.

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We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity, and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation, and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.

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Dias, Claudio Ferreira, Felipe A. P. de Figueiredo, Eduardo Rodrigues de Lima, and Gustavo Fraidenraich. "Sum-Rate Channel Capacity for Line-of-Sight Models." Sensors 21, no. 5 (March 1, 2021): 1674. http://dx.doi.org/10.3390/s21051674.

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This work considers a base station equipped with an M-antenna uniform linear array and L users under line-of-sight conditions. As a result, one can derive an exact series expansion necessary to calculate the mean sum-rate channel capacity. This scenario leads to a mathematical problem where the joint probability density function (JPDF) of the eigenvalues of a Vandermonde matrix WWH are necessary, where W is the channel matrix. However, differently from the channel Rayleigh distributed, this joint PDF is not known in the literature. To circumvent this problem, we employ Taylor’s series expansion and present a result where the moments of mn are computed. To calculate this quantity, we resort to the integer partition theory and present an exact expression for mn. Furthermore, we also find an upper bound for the mean sum-rate capacity through Jensen’s inequality. All the results were validated by Monte Carlo numerical simulation.

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37

HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (July 2, 2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.

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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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LOH, PO-SHEN, and BENNY SUDAKOV. "On the Strong Chromatic Number of Random Graphs." Combinatorics, Probability and Computing 17, no. 2 (March 2008): 271–86. http://dx.doi.org/10.1017/s0963548307008607.

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Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding $k \big\lceil \frac{n}{k} \big\rceil - n$ isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

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39

Zhu, Jianshen, Chenxi Wang, Aleksandar Shurbevski, Hiroshi Nagamochi, and Tatsuya Akutsu. "A Novel Method for Inference of Chemical Compounds of Cycle Index Two with Desired Properties Based on Artificial Neural Networks and Integer Programming." Algorithms 13, no. 5 (May 18, 2020): 124. http://dx.doi.org/10.3390/a13050124.

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Inference of chemical compounds with desired properties is important for drug design, chemo-informatics, and bioinformatics, to which various algorithmic and machine learning techniques have been applied. Recently, a novel method has been proposed for this inference problem using both artificial neural networks (ANN) and mixed integer linear programming (MILP). This method consists of the training phase and the inverse prediction phase. In the training phase, an ANN is trained so that the output of the ANN takes a value nearly equal to a given chemical property for each sample. In the inverse prediction phase, a chemical structure is inferred using MILP and enumeration so that the structure can have a desired output value for the trained ANN. However, the framework has been applied only to the case of acyclic and monocyclic chemical compounds so far. In this paper, we significantly extend the framework and present a new method for the inference problem for rank-2 chemical compounds (chemical graphs with cycle index 2). The results of computational experiments using such chemical properties as octanol/water partition coefficient, melting point, and boiling point suggest that the proposed method is much more useful than the previous method.

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BABARO, JUAN PABLO, and GASTON GIRIBET. "ON THE DESCRIPTION OF SURFACE OPERATORS IN ${\mathcal N} = 2^*$ SYM." Modern Physics Letters A 28, no. 06 (February 22, 2013): 1330003. http://dx.doi.org/10.1142/s0217732313300036.

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Alday and Tachikawa [Lett. Math. Phys.94, 87 (2010)] observed that the Nekrasov partition function of [Formula: see text] superconformal gauge theories in the presence of fundamental surface operators can be associated to conformal blocks of a 2D CFT with affine sl(2) symmetry. This can be interpreted as the insertion of a fundamental surface operator changing the conformal symmetry from the Virasoro symmetry discovered in Ref. 2 to the affine Kac–Moody symmetry. A natural question arises as to how such a 2D CFT description can be extended to the case of non-fundamental surface operators. Motivated by this question, we review the results [Y. Hikida and V. Schomerus, JHEP0710, 064 (2007); S. Ribault, JHEP0805, 073 (2008)] and put them together to suggest a way to address the problem: It follows from this analysis that the expectation value of a non-fundamental surface operator in the SU(2) [Formula: see text] super Yang–Mills (YM) theory would be in correspondence with the expectation value of a single vertex operator in a two-dimensional CFT with reduced affine symmetry and whose central charge is parametrized by the integer number that labels the type of singularity of the surface operator.

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Knopfmacher, Arnold, and Augustine O. Munagi. "Successions in integer partitions." Ramanujan Journal 18, no. 3 (December 30, 2008): 239–55. http://dx.doi.org/10.1007/s11139-008-9140-2.

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MUTAFCHIEV, LJUBEN. "The Size of the Largest Part of Random Weighted Partitions of Large Integers." Combinatorics, Probability and Computing 22, no. 3 (February 21, 2013): 433–54. http://dx.doi.org/10.1017/s0963548313000047.

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We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of conditions, we prove thatXn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equalse−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

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43

Oruç, A. Yavuz. "On number of partitions of an integer into a fixed number of positive integers." Journal of Number Theory 159 (February 2016): 355–69. http://dx.doi.org/10.1016/j.jnt.2015.06.023.

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GOYAL, NAVIN, LUIS RADEMACHER, and SANTOSH VEMPALA. "Query Complexity of Sampling and Small Geometric Partitions." Combinatorics, Probability and Computing 24, no. 5 (October 21, 2014): 733–53. http://dx.doi.org/10.1017/s0963548314000704.

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In this paper we study the following problem.Discrete partitioning problem (DPP). Let$\mathbb{F}_q$Pndenote then-dimensional finite projective space over$\mathbb{F}_q$. For positive integerk⩽n, let {Ai}i= 1Nbe a partition of ($\mathbb{F}_q$Pn)ksuch that:(1)for alli⩽N,Ai= ∏j=1kAji(partition into product sets),(2)for alli⩽N, there is a (k− 1)-dimensional subspaceLi⊆$\mathbb{F}_q$Pnsuch thatAi⊆ (Li)k.What is the minimum value ofNas a function ofq, n, k? We will be mainly interested in the casek=n.DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: when the dimension is low (k=n= 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.

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MARTIN, PAUL. "TEMPERLEY-LIEB ALGEBRAS FOR NON-PLANAR STATISTICAL MECHANICS — THE PARTITION ALGEBRA CONSTRUCTION." Journal of Knot Theory and Its Ramifications 03, no. 01 (March 1994): 51–82. http://dx.doi.org/10.1142/s0218216594000071.

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We give the definition of the Partition Algebra Pn(Q). This is a new generalisation of the Temperley–Lieb algebra for Q-state n-site Potts models, underpinning their transfer matrix formulation on arbitrary transverse lattices. In Pn(Q) subalgebras appropriate for building the transfer matrices for all transverse lattice shapes (e.g. cubic) occur. For [Formula: see text] the Partition algebra manifests either a semi-simple generic structure or is one of a discrete set of exceptional cases. We determine the Q-generic and Q-independent structure and representation theory. In all cases (except Q = 0) simple modules are indexed by the integers j ≤ n and by the partitions λ ˫ j. Physically they may be associated, at least for sufficiently small j, to 2j 'spin' correlation functions. We exhibit a subalgebra isomorphic to the Brauer algebra.

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Canfield, E. Rodney, Carla D. Savage, and Herbert S. Wilf. "Regularly spaced subsums of integer partitions." Acta Arithmetica 115, no. 3 (2004): 205–16. http://dx.doi.org/10.4064/aa115-3-1.

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Archibald, Margaret, Aubrey Blecher, and Arnold Knopfmacher. "Distinct r-tuples in integer partitions." Ramanujan Journal 50, no. 2 (September 20, 2019): 237–52. http://dx.doi.org/10.1007/s11139-019-00180-x.

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Leader, Imre, and Paul A. Russell. "Inhomogeneous Partition Regularity." Electronic Journal of Combinatorics 27, no. 2 (June 26, 2020). http://dx.doi.org/10.37236/7972.

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We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved that the system $Ax = b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general (commutative) ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new 'direct' proof of Rado’s result.

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Dilcher, Karl, and Larry Ericksen. "Polynomial analogues of restricted multicolor b-ary partition functions." International Journal of Number Theory, June 16, 2020, 1–21. http://dx.doi.org/10.1142/s1793042120400096.

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Given an integer base [Formula: see text], a number [Formula: see text] of colors, and a finite sequence [Formula: see text] of positive integers, we introduce the concept of a [Formula: see text]-restricted [Formula: see text]-colored [Formula: see text]-ary partition of an integer [Formula: see text]. We also define a sequence of polynomials in [Formula: see text] variables, and prove that the [Formula: see text]th polynomial characterizes all [Formula: see text]-restricted [Formula: see text]-colored [Formula: see text]-ary partitions of [Formula: see text]. In the process, we define a recurrence relation for the polynomials in question, obtain explicit formulas, and identify a factorization theorem.

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Schneider, Robert, and Andrew V. Sills. "Analysis and combinatorics of partition zeta functions." International Journal of Number Theory, February 21, 2020, 1–10. http://dx.doi.org/10.1142/s1793042120400023.

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We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

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